= Force-extension of an intrinsically curved filament
{title2=$\widehat\zeta(q)=-B\widehat\kappa_0(q)/(F+Bq^2)$}
For a planar <elastic filament> under positive constant extension force $F$, the quadratic force-controlled energy is
$$
\mathcal H=\frac12\int\left[B(\zeta''-\kappa_0)^2+F(\zeta')^2\right]dx.
$$
The <higher-order Euler-Lagrange equation> is $B\zeta''''-F\zeta''=B\kappa_0''$. On a periodic interval or in a bulk <Fourier transform> treatment, every nonzero mode obeys the displayed transfer relation. The length deficit is $\frac12\int(\zeta')^2dx$, so its ensemble value is determined by the <power spectrum> of the <quenched intrinsic curvature>. Boundary conditions must be specified for a finite filament.
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